Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-3x+3y &= -1 \\ 6x+6y &= -9\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}6x-6y &= 2\\ 6x+6y &= -9\end{align*}$ Add the top and bottom equations. $12x = -7$ Divide both sides by $12$ and reduce as necessary. $x = -\dfrac{7}{12}$ Substitute $-\dfrac{7}{12}$ for $x$ in the top equation. $-3( -\dfrac{7}{12})+3y = -1$ $\dfrac{7}{4}+3y = -1$ $3y = -\dfrac{11}{4}$ $y = -\dfrac{11}{12}$ The solution is $\enspace x = -\dfrac{7}{12}, \enspace y = -\dfrac{11}{12}$.